Volumen 26 (2005), p´aginas 145–163
Spectral classification in a periodic structure
1 F. GheribCentre universitaire de Mascara, Alg´
erie
B. MessirdiUniversit´
e d’ Oran Es-senia, Alg´
erie
Abstract. In this paper we consider the Schr¨odinger operators
P = −d2
dx2 + V , where V is a periodic real function. Essentially we show that the spectra of these operators can be determined by a smooth function α, the phase of the eigenvalues of a matrix M (E, x, V ) express-ing the states of a particle in a L− periodical structure. The recurrent spectrum and the transient spectrum of P are also calculated. Such structures are met meet, for example, in solid state physics in the study of a linear molecule formed of regularly spaced atoms.
Key words and phrases. Schr¨odinger operator, periodic potential,
ab-solutely continuous spectrum, transient and recurrent spectrum.
2000 AMS Classification Subject: 35P15, 35Q20, 35P99, 47A53. Resumen.En este trabajo se estudian los operadores de Schr¨odinger
P = −d2
dx2 + V , donde V es una funci´on peri´odica real. Esencialmente, mostramos que los espectros de estos operadores pueden determinarse con mediante una funci´on suave α, que es la fase de los valores propios de una matriz M (E, x, V ) que expresa los estados de una part´ıcula en una estructura L−peri´odica. Tambi´en calculamos los espectos recur-rente y transiente de P . Estas estructuras se encuentran, por ejemplo, al estudiar en la f´ıca del estado s´olido una mol´ecula lineal formada por ´
atomos regularmente espaciados.
1Investigation supported by University of Oran Es-senia, Algeria. Research topics CNEPRU.
1. Introduction
We know that the spectral properties of Schr¨odinger operators P =− d2
dx2+ V on L2(R) depend mainly on the behavior of the real potential function V at infinity.
Basically, in spectral theory we distinguish four distinct classes of Schr¨ odin-ger operators P. In the first class V (x) → ∞ as |x| → +∞, and has empty essential spectrum. The next simplest class consists of Schr¨odinger operators for which V (x) → 0 as |x| → +∞. Under fairly general hypotheses these operators have σess(P ) = [0, +∞[ for essential spectrum and empty singular continuous spectrum σsc(P ) = ∅. The essential tools used to establish these results are the min-max principle, Weyl’s theorem, perturbation theory of linear operators, etc. (See [ReSi, theorems xiii15, xiii16, and xiii33]; [CyFrKiSi], [Ka] and [DaLi].)
The third class is made up of the N−body Schr¨odinger operators for which
V (x) → 0 as |x| → +∞ in most directions (e.g., Coulombic interactions).
Under specialized hypotheses, for these operators we have σess(P ) = [α, +∞[
and σsc(P ) =∅, where α ≥ 0. (See [ReSi, theorems xiii17 and xiii36].)
In the fourth and last class one finds Schr¨odinger operators P where V is a periodic real function. These operators, considered as unbounded and self adjoint on L2(R), are important in solid state physics. Indeed, periodicity is the property that characterizes the potential to which a particle in a crystal is submitted.
If V is L−periodic it has not a limit as |x| → +∞ , so the spectral analysis of periodic Schr¨odinger operators is difficult (see e.g. [AvSi2], [Ea]). Nevertheless, the property that allows the study of these situations is that P has a large symmetry group
[P, U (L)] = 0 ,
where U (L) is the unitary operator defined on L2(R) by U(L)ϕ(x) = ϕ(x + L), and [P, U (L)] is the commutator of P and U (L).
It is known that (see eg. [AvSi2], [Ea], [DaLi]) the study of these operators has been based on the concept of decomposable operators by the direct integral decomposition so far. Also, the classification of the spectrum of these opera-tors is done naturally studying the spectrum of their fibers. It follows that Schr¨odinger operators with sufficiently smooth potentials and 2π-periodical have purely absolutely continuous spectrum and are unitarily equivalent to the
Hilbertian integral operator ⊕ [0,2π] P (θ)dθ 2π,
where the fiber P (θ) is the operator P =− d
2 dx2+V defined on L 2 [0, 2π],dθ 2π , with boundary conditions:
ψ(2π) = eiθψ(0) , ψ(2π) = eiθψ(0) .
For all θ∈ [0, 2π], P (θ) has a compact resolvent and a discrete spectrum, which reveals the existence in the spectrum of P continuous bands of energy known as allowed bands, possibly separated by energy bands known as forbidden bands. The spectrum of P on L2(R) is the union of all the allowed bands.
Several recent papers have treated the localization of bands of energy and the estimation of their width. For this we refer to results obtained by Outassourt in the semi-classical case [Ou].
The main difficulties appear while extending the analysis of the fibers of
P . For this analysis depends critically on the simplicity of the eigenvalues of P (θ), simplicity which fails in the multidimentional case since eigenvalues are
not necessarily analytic at degeneracy points [Me].
Here we plan to give a new and original spectral analysis of the Schr¨odinger operators with real and continuous L−periodical potentials. Our method is also valid in the degenerate case and will allow us to obtain all the quoted results by using direct and easy proofs. In particular, we obtain in this way a precise refinement of the absolutely continuous spectrum of P .
Our technique is based on the behavior of the eigenfunctions of P at the time of the crossing of the totality of the successive barriers created by the periodic potential V . For this, we introduce for E real a matrix of transi-tion M (E, x, V ) with eigenvalues e± w(E)L such that w(E) = γ(E) + iα(E).
M (E, x, V ) describes the states of a particle in a L−periodical structure V at
the energy E and the position x.
The function α is known as the phase function. We show that α is differen-tiable and obtain our main result
σ(P ) =
E∈ R ; dα
dE(E)= 0
(Here σ stands for the spectrum). The function γ(E) characterizes the spec-trum of the fibers of P and its resolvent set and γ(E) vanishes on the allowed bands.
Using the function α it is now possible in this case to deduce the spectral measure and the various components of the absolutely continuous spectrum: the transient spectrum σrac(P ) and the recurrent spectrum σtac(P ) of P . This decomposition is possible thanks to a partition of the positive density of the spectral measure of P in two essential parts: the essential interior and the essential frontier (see. [GhMe],[AvSi1]).
2. Behavior of the solutions in a periodic structure
Consider
− d2
dx2 + (V (x)− E)
ϕ = 0 (E)
where V is a real continuous function onR which is constant in the exterior of the compact [0 , L]:
V (x) = V0, ∀x ∈ R [0 , L] ,
(V0= V (0) = V (L)) , L, E∈ R , such that V0< E .
Let ϕ1(E, x) and ϕ2(E, x) be two linearly independant solutions of (E), then
ϕi(E, x) , i = 1, 2, is analytic in E and C2 for each x by the standard theory
of ordinary differential equations. In particular,
ϕ1(E, x) =
eikx if x≤ 0
F (E)eikx+ G(E)e−ikx if x≥ L
ϕ2(E, x) =
e−ikx if x≤ 0
F(E)eikx+ G(E)e−ikx if x≥ L
with k = √E− V0. F (E), G(E), F(E) and G(E) are coefficients depending on V and analytically on E.
Since V is real, we see that if ϕ is solution of (E) its conjugated is also solution. One can take therefore F(E) = G(E) and G(E) = F (E).
All other solutions of (E) are of the form
Thus ϕ(E, x) = A1eikx+ A1e−ikx if x≤ 0 A1eikx+ A 1e−ikx if x≥ L (2.1) and A1= F (E)A1+ G(E)A1 A1= G(E)A1+ F (E)A1 , (2.2) where A1, A1, A1, and A1 are some complex constants. The coefficients F (E) and G(E) allow us to know the behavior of the solution ϕ on the left of the potential from its behavior on the right of V .
We now want to study the behavior of the solution at the time of the crossing of all the successive barriers, that is to find the solution ϕ(E, x) of the equation (E) where V is a L−periodical superposition on R of functions of the previous type.
By virtue of the previous calculus, the general solution of (E) is given on R by: ϕ(E, x) = A1ϕ1(E, x) + A1ϕ2(E, x) if 0≤ x ≤ L A2ϕ1(E, x− L) + A2ϕ2(E, x− L) if L ≤ x ≤ 2L · · · An+1ϕ1(E, x− nL) + An+1ϕ2(E, x− nL) if nL≤ x ≤ (n + 1)L , n ∈ Z
Moreover, it is easy to calculate the values of ϕ1(E, x) and ϕ1(E, x), as well as those of their derivatives at the extremities of every period. So, ϕ(E, x) has the same value and the same derivative than the function
Aneik(x−(n−1)L)+ Ane−ik(x−(n−1)L)
at the extremity on the left of the n−th period. In the same way, on the right, it has the same value and the same derivative than
Aneik(x−(n−1)L)+ Ane−ik(x−(n−1)L).
Then,
Aneik(nL−(n−1)L)+ Ane−ik(nL−(n−1)L)= An+1eik(nL−nL)+ An+1e−ik(nL−nL)
and
ik[ Aneik(nL−(n−1)L)− Ane−ik(nL−(n−1)L)]
and therefore, for all n inN∗, An+1= eikLA n An+1= e−ikLA n . Using (2.2), we obtain An+1 An+1 = (Q(E))n A1 A1 , ∀n ∈ N∗, (2.3) where Q(E) =
eikLF (E) eikLG(E)
e−ikLG(E) e−ikLF (E)
. (2.4)
Now the characteristic equation of Q(E) is
λ2− D(E)λ + 1 = 0 ,
where D(E) = T r(Q(E)) = eikLF (E) + e−ikLF (E) = 2ReeikLF (E).
The calculus of (Q(E))nwill be easy if one changes the basis to get a diagonal form of Q(E). This is the reason why we are going to study the spectrum of
Q(E).
Then, the eigenvalues of Q(E) can be expressed in the form λ±(E) =
e± w(E)L with w(E)=γ(E) + iα(E), γ(E)≥ 0; such as:
Propisition 2.1.
(i) γ(E) = 0⇔ |D(E)| ≤ 2 (ii)|D(E)| > 2 ⇒
α(E) = π/L if D(E) > 0 α(E) = 0 if D(E) < 0
Proof. We only prove (ii). Indeed, for |D(E)| > 2, the eigenvalues of Q(E) are λ±(E) = εe±γ(E)L, ε =±1.
If
ε =−1, D(E) < 0 , λ±(E) = e± [γ(E)+i(π/L)]L, then α(E) = π L.
If
3. Original calculus of the continuous spectrum
By virtue of Proposition 2.1, we note that there are two types of energy E. The energy E such that|D(E)| ≤ 2 or |D(E)| > 2.
We show that: σ(P ) ={E ∈ R ; |D(E)| ≤ 2} = E∈ R ; dα dE(E)= 0
for that we are brought more particularly to study the properties of the function
α(E) when V is a real continuous function and periodic onR.
Let ϕ1(E, x) and ϕ2(E, x) be two independent solutions of the equation (E), such that ϕ1(E, 0) = 1 dϕ1 dx (E, 0) = 0 and ϕ2(E, 0) = 0 dϕ2 dx (E, 0) = 1
Consider the matrix M (E, x, V ) defined by
M (E, x, V ) = ϕ1(E, x) ϕ2(E, x) dϕ1 dx (E, x) dϕ2 dx (E, x) (3.1)
then by using (2.1) and (2.2) we have
M (E, L, V ) =
ReeikLF (E)+ Ree−ikLG(E) −kImeikLF (E)+ kIme−ikLG(E)
1 kIm eikLF (E)+1 kIm e−ikLG(E)
ReeikLF (E)− Ree−ikLG(E)
We observe that T r(M (E, L, V )) = T r(Q(E)) = D(E), thus M (E, L, V ) and Q(E) have the same characteristic equation and therefore the same eigen-values.
Consequently, we obtain the following corollary:
Corollary 3.1. The following assertions are equivalent:
(1) |D(E)| ≤ 2 (2) γ(E) = 0
(4) eiα(E)L is an eigenvalue for M (E, L, V ) .
Let us consider P (θ) the operator P defined on L2([0, L]) with the boundary
conditions
ψ(L) = eiθψ(0)
ψ(L) = ei θψ(0) where θ∈ [0, 2π]. Then,
Theorem 3.2. γ(E) = 0 if and only if there exist θ∈ [0, π] , such that E is an eigenvalue for P (θ). In this case, we have:
θ = α(E)L
Proof. If ϕ is a solution of (E), then
ϕ(E, x) = c1ϕ1(E, x) + c2ϕ2(E, x) . Thus,
ϕ(E, L) = c1ϕ1(E, L) + c2ϕ2(E, L)
dϕ dx(E, L) = c1 dϕ1 dx (E, L) + c2 dϕ2 dx (E, L) ϕ(E, 0) = c1 dϕ dx(E, 0) = c2 and then ϕ(E, L) dϕ dx(E, L) = M(E, L, V ) ϕ(E, 0) dϕ dx(E, 0) .
If eiα(E)L is an eigenvalue for M (E, L, V ), there exist c1 and c2 such that (c1, c2)= (0, 0) and M (E, L, V ) c1 c2 = eiα(E)L c1 c2 . We put
ψ(E, x) = c1ϕ1(E, x) + c2ϕ2(E, x) .
ψ is thus an eigenfunction associated with the eigenvalue E of the operator P (α(E)L). Conversely, if E is an eigenvalue for P (θ) and θ∈ [0, π] , then there
exist Ψ(E, .)= 0 solution of (E) such that: Ψ(E, L) dΨ dx(E, L) = eiθ Ψ(E, 0) dΨ dx(E, 0) = M(E, L, V ) Ψ(E, 0) dΨ dx(E, 0) , which shows that eiθ is an eigenvalue for M (E, L, V ) and θ = α(E)L.
We are thus brought to study the operators P (θ), θ∈ [0, π]. We recall some properties of these operators.
Theorem 3.3. [See. [GhMe],[ReSi]]
(i) for all θ∈ [0, π], P (θ) has purely discrete spectrum (En(θ))n∈N∗, such
that for any n, En(.) is analytic on ]0, π[ and continuous at 0 and π.
(ii) P (θ) and P (2π−θ) are antiunitarily equivalent under ordinary complex
conjugation. In particular, their eigenvalues are identical and their eigenfunctions are complex conjugates.
(iii For n odd (respectively, n even) En(.) is strictly increasing
(respec-tively, strictly decreasing) as θ increases from 0 to π. In particular,
0 < E1(0) < E1(π)≤ E2(π) < E2(0)≤ · · ·
≤ E2n−1(0) < E2n−1(π)≤ E2n(π) < E2n(0)≤ · · ·
We can now combine Corollary 3.1 and Theorems 3.2 and 3.3 to get:
Theorem 3.4. If P =− d2 dx2+ V is now defined on L 2(R) with domain H2(R) = u∈ L2(R) ;d 2u dx2 ∈ L 2(R)
and V is a continuous real and periodic function onR, we have {E ∈ R ; γ(E) = 0} =
n∈N∗
θ∈[0,π]
{En(θ)}
and, if ρ designates the resolvent set,
{E ; γ(E) > 0} ⊂ ρ(P ) .
Proof. Indeed, if γ(E) > 0, by the standard theory of ordinary differential
equations, we obtain two linearly independent functions ϕ1(E, .) and ϕ2(E, .) analytic in E and C2(R) solutions of the equation (E) such that
where v1(E, x) and v2(E, x) are two periodic functions. Moreover, for all x∈ R:
ϕ1(E, x)dϕ2
dx (E, x)− ϕ2(E, x) dϕ1
dx (E, x) = constant=c= 0 . (3.2)
Let us define the function
GE(x, y) = GE(y, x) =−
ϕ1(E, x)ϕ2(E, y)
c , x≤ y ,
and consider the integral operator GE with kernel GE(x, y) acting on L2(R)
by (GEf )(x) = R GE(x, y)f (y)dy =−1 c[ϕ2(E, x) x −∞
ϕ1(E, y)f (y)dy + ϕ1(E, x) +∞
x
ϕ2(E, y)f (y)dy]. (3.3)
Now GE is a bounded operator on L2(R). Indeed, since there exist k1, k2> 0
such that for all x inR,
|ϕ1(E, x)| ≤ k1eγ(E)x ; |ϕ2(E, x)| ≤ k2e−γ(E)x,
it suffices to establish that the operators
K−f (x) = x −∞ e−γ(E)(x−y)|f(y)| dy ; K+f (x) = +∞ x e−γ(E)(y−x)|f(y)| dy map continuously on L2(R).
In particular, we see that K± are well defined on L2(R). By partial integra-tion and Schwarz’ inequality we see that, for any T > 0,
T −T |K−f (x)|2dx = T −T e−2γ(E)x[ x −∞
eγ(E)y|f(y)| dy]2dx
= − 1 2γ(E)e −2γ(E)x x −∞ eγ(E)y|f(y)| dy 2 x=T x=−T + 1 γ(E) T −T e−γ(E)x|f(x)| x −∞ eγ(E)y|f(y)| dy dx ≤ 1 2γ(E)(K−f (−T )) 2+ 1 γ(E) T −T |K−f (x)| |f(x)| dx ≤ 1 2γ(E)(K−f (−T )) 2 + 1 γ(E) T −T |K−f (x)|2dx 1/2 T −T |f(x)|2dx 1/2 . However, lim T →+∞K−f (−T ) = 0, so that K−f L2(R)≤ 1 γ(E) f L2(R) .
We obtain the same estimation for K+. Consequently, for arbitrary f , g ∈
L2(R), we have
K±f− K±g L2(R)≤
1
γ(E) f − g L2(R) . (3.4)
Then the K± are continuous. This is so since GE is the inverse operator of (P − E) and then E ∈ ρ(P ). Indeed,
(i) For all f in H2(R) we have (by using derivatives in the generalized sense) (P − E)GEf (x) =−1 c −d2ϕ2 dx2 (E, x) x −∞
ϕ1(E, y)f (y)dy−
dϕ2 dx (E, x)ϕ1(E, x)f (x) + dϕ1 dx (E, x)ϕ2(E, x)f (x)− d2ϕ1 dx2 (E, x) +∞ x
ϕ2(E, y)f (y)dy+
(V (x)− E)ϕ2(E, x)
x
−∞
ϕ1(E, y)f (y)dy+
(V (x)− E)ϕ1(E, x)
+∞
x
ϕ2(E, y)f (y)dy = f(x) (ii) After two partial integrations we obtain
(GE(P− E)f)(x) = +∞ −∞ GE(x, y)(P − E)f(y)dy =−1 c ϕ2(E, x)dϕ1 dx (E, x)− ϕ1(E, x) dϕ2 dx (E, x) f (x)+
(GE(P − E)ϕ1(E, x)f )(x) + (GE(P − E)ϕ2(E, x)f )(x) = f (x)
for all f ∈ H2(R)
Now, consider the bijective mappings
En: [0, π]→ [En(0), En(π)] , θ→ En(θ) if n is odd, and
En: [0, π]→ [En(π), En(0)] θ→ En(θ) if n is even.
If for instance n is odd and E ∈ [En(0), En(π)] then there is a unique
θ∈ [0, π] such that E = En(θ), and eiθis an eigenvalue for M (E, L, V ).
obtain that
α( . )L : [En(0), En(π)]→ [0, π] , E → α(E)L , is the inverse mapping of
En: [0, π]→ [En(0), En(π)] , θ→ En(θ) .
It is then continuous and strictly monotone, and thus strictly increasing on ]0, π[, because En is analytic
α(En(0))L = 0
α(En(π))L = π .
Therefore, α is differentiable on the interval ]En(0), En(π)[. By the same
method, we show that for n even the application
α( . )L : [En(π) , En(0)]→ [0, π] , E → α(E)L ,
is the inverse of
En: [0, π]→ [En(π), En(0)] , θ→ En(θ) .
It is continuous and strictly decreasing on ] 0 , π[, so it is differentiable on this interval. Put αn= En(0) if n is odd En(π) if n is even , βn= En(π) if n is odd En(0) if n is even
We have obtained the following result:
Lemma 3.5. The phase α is possibly constant on the intervals ] βn, αn+1[. Or
it is strictly increasing for n odd and strictly decreasing for n even on ] αn, βn[.
Let f (E) = 0 if E∈ ] βn, αn+1[ (−1)n−1dα dE(E) if E∈ ] αn, βn[ (3.5)
According to Lemma 3.5, f is dE−almost everywhere defined on R. More-over it is locally integrable because it is bounded, so the corresponding abso-lutely continuous measure dν = f (E)dE is then well defined (where dE is the Lebesgue measure inR denoted by | . | in section 4).
Theorem 3.6. If V is a continuous real and periodic function onR , then the spectrum of the operator P on L2(R) is purely absolutely continuous.
Proof. Let us define for all n in N∗ Hn= Ψ∈ L2(R, dx) ; ∃g ∈ L2 [0, 2π] ,dθ 2π , Ψ(x + kL) = 2π 0 eikθg(θ)Ψn(θ, x)dθ 2π, 0≤ x ≤ L, k ∈ Z
where{Ψn(θ, .)}n∈N∗ is an orthonormal basis of eigenfunctions associated with the discrete spectrum (En(θ))n∈N∗ of P (θ). Hn is a closed subspace of
L2(R, dx) and L2(R, dx) = n=+∞
n=1 Hn
. On the other hand, P preserves all the spacesHn, n∈ N∗. Indeed, for all Ψ∈ Hn∩ H2(R), we have:
(P Ψ)(x + kL) = 2π 0 eikθg(θ)En(θ)Ψn(θ, x)dθ 2π ; ∀k ∈ Z
and since En( . ) is analytic on [0, 2π] then g( . )En( . )∈ L2
[0, 2π] ,dθ 2π . Con-sequently, σ(P ) = n=+∞ n=1 σ(P|Hn) (3.6)
In particular, if we set for all n inN∗
Un:Hn → L2 [0, 2π] ,dθ 2π , Ψ→ g such that Ψ(x + jL) = 2π 0 eijθg(θ)Ψn(θ, x) dθ 2π,
we then see that U\ is unitary. We get Pn = UnPUn−1, then Pn(g( . )) =
En( . )g( . ), for all n∈ N∗. Furthermore,
L2 [0, 2π] ,dθ 2π = L2 [0, π] ,dθ 2π L2 [π, 2π] , dθ 2π and Pn preserves these two spaces. If for instance n is odd,
α is a C1function and dα is absolutely continuous with respect to the Lebesgue measure.
For the operator U from L2
[0, π] ,dθ 2π
into L2([En(0), En(π)] , dx),
de-fined by (Ug)(E) = dα dE 1/2 g(α(E)) = ! dEn dθ θ=α(E) "−1/2 g(α(E)) , we have (Ug) 2L2([En(0), En(π)],dx) = En(π) En(0) |g(α(E))|2 (dEn dθ )θ=α(E) −1 dE = π 0 |g(x)|2dx
We see thatU is a surjective isometry, i.e. unitary. Moreover,
UPnU−1
g(E) = Eg(E) ; (3.7)
then each Pn has an absolutely continuous spectrum, and so does P according (3.6).
Theorem 3.7. If V is a continuous real and periodic function onR and µ the spectral measure associated with the operator P, then µ is equivalent to ν.
In order to show this theorem we are going to need the following fundamental results:
Lemma 3.7. [ See. [Is],[Pa],[JoMo], [Ko],[Ri]]
(1) In the general case γ can be extended to continuous potentials, not
necessarly periodic, using the expression γ(E) = lim
|x|→+∞
1
|x|log M(E, x, V ) . (3.8)
(2) Let dµac(E) = g(E)dE the absolutely continuous part of the spectral
measure µ with respect to the Lebesgue measure dE, then for all Borel sets B ofR, we have γ(E) = 0 , dE−almost everywhere on B ⇒ g(E) =
Proof of theorem 3.7 Let B be a Borel set of R. Then, we have ν(B) = 0⇒ f(E) = 0 , dE- almost everywhere on B
⇒ dα
dE(E) = 0 , dE- almost everywhere on B ⇒ γ(E) > 0 , dE- almost everywhere on B .
Now thanks to Lemma 3.4, we know that E∈ ρ(P ) if γ(E) > 0. Thus
ν(B) = 0⇒ E ∈ ρ(E) ⇒ µac(B) = 0 .
Moreover, ν(B)= 0 ⇒ f(E) = 0, dE−almost everywhere on B ⇒ dα
dE(E)= 0, dE−almost everywhere on B ⇒ γ(E) = 0.
According to Lemma 3.8, we also have γ(E) = 0, dE−almost everywhere on
B ⇒ f(E) > 0, dE−almost everywhere on B. Then ν(B) = 0 ⇒ µac(B)= 0, and σac(P ) = supp µac= suppf = E∈ R ; dα dE(E)= 0 ={E ∈ R ; γ(E) = 0} According to Lemma 3-4, σac(P ) = σ(P ) ={E ∈ R ; γ(E) = 0} = +∞ n=1 [αn, βn]
and dν(E) = dµac(E) = f (E)dE.
4. Decomposition of the absolutely continuous spectrum
In this section we give a refinement of absolutely continuous spectrum of the operator P on L2(R), when V is continuous real and periodic function on R.
Now, using results from the articles of Avron-Simon [AvSi1] and Gherib-Messirdi [GhMe], it is easy to establish that σ(P ) is decomposed into two disjoint sets called respectivelly the transient σtac(P ) and reccurent σrac(P )
absolutely continuous spectrum of P . Set
B =
+∞
n=1
For that it is enough to determine the event #oes
B
$
of the essential interior of the [B] and the event [C] of the essential frontier of the event [B]. Where the event of the borelian set B is
[B] =%A borelian set ofR ;&&A∆B&&= 0'; its essential interior notedoesB is defined by
%
x∈ R ; ∃ t > 0 such that&&]x− t, x + t[ ∩ B&&= 2t'
and the essential frontier of B is the event [C], where C is a borelian set of R satisfying &&&C∆(BoesB )&&& = 0, ∆ denotes the symmetric difference between
subsets ofR.
Since the essential interior contains the topological interior, then
+∞
n=1
]αn, βn[⊂ oes
B .
In the other hand if αn is a degenerate point for P (π) if n is even and for P (0)
if n is odd, then there exists t > 0 such that : && &&]αn− t , αn+ t[ ∩ σ(P ) && && = 2t Thus αn∈ oes
B . Now, if αn is a nondegenerate point for P (π), if n is even (n is
taken to be odd for the case P (0)), then for all t > 0: &&
&&]αn− t , αn+ t[ ∩ σ(P )
&& && < 2t
and αnmust belong to C. In the other hand, if βn is a nondegenerate point for
P (π), if n is odd (n is taken to be even for the case P (0)), then for all t > 0:
&&
&&]βn− t , βn+ t[ ∩ σ(P )
&& && < 2t Consequently, βn must belong to C and
oes
B =
+∞
n=1
]αn, βn[∪ {αn; αnis a degenerate point for
C ={αn ; αn is a nondegenerate point forP (π)if n is even
and for P (0) if n is odd}
∪ {βn ; βn is a nondegenerate point for P (0) if n is even
and for P (π) if n is odd} Since
L2tac= p(oesB )L2ac and L2rac= p (C) L2ac (4.2)
where p (oesB ) and p (C) are respectively the orthogonal projections on oesB and C and L2ac is the absolutely continuous space associated with the operator P on L2(R), then σtac(P ) = σ ( P|L2 tac ) ⊂oesB and σrac(P ) = σ P|L2 rac ⊂ C
In additionoesB and C are a partition of σac, then
σtac(P ) = oes
B and σrac(P ) = C (4.3)
References
[AvSi1] J. Avron, B. Simon. Transient and reccurent spectrum. J. Func. Anal. 43 (1981), 1–31.
[AvSi2] J. Avron, B. Simon. Almost periodic Schr¨odinger operators, I. Limit periodic potentials. Commun. Math. Phys.obf 82 (1982), 101–120.
[CyFrKiSi] H. L. Cycon, R. G. Froese, W. Kirsh, B. Simon. Schr¨odinger operators, with applications to quantum mechanics and global geometry. Springer–Verlag,
New York, 1986.
[DaLi] R. Dautry, J. -L. Lions. Analyse math´ematique et calcul num´erique. Spectre
des op´erateurs (tome 5). Masson, Paris, 1988.
[Ea] M. S. P. Eastham. The spectral theory of periodic differential equations.
Hainer, New York, 1974.
[GhMe] F. Gherib, B. Messirdi. Various classification of the continuous spectrum of
a selft-adjoint operator. Research topics CNEPRU, B3101/13/00, 2002.
[Is] K. Ishii. Localization of eigenstates and transport phenomena in
one-dimentional disordered systems. Suppl. Prog. theor. phys.53, 1973, 77–183.
[JoMo] R. Johnson, J. Moser. The rotation number for almost periodic potentials.
Commun. Math. Phys. Erratum, Commun. Math.84, (1982), 403–438. [Ka] T. Kato. Perturbation theory for linear operators. Springer–Verlag, Heidelberg,
New York, 1976.
[Ko] S. Kotani. Ljaponov indices determine absolutely continuous spectra of
Stocastic Analysis, Katata(1982), 225–247. Edited by K. Ito. North Holland,
Ameterdam, 1984.
[Me] B. Messirdi. Asymptotique de Born-Oppenheimer pour la pr´edissociation mol´eulaire(cas de potentiels r´eguliers). Annales de l’Institut Henri Poincare,61
(1994), 255–292.
[Ou] A. Outassourt. Comportement semi-classique pour l’op´erateur de Schr¨odinger `a potentiel p´eriodique. Journal of Functional Analysis72 (1987),
65–93.
[Pa] L. Pastur. Spectral theory of random selft-adjoint operators”. Journal of soviet
Mathematics46 (1989), xxx.
[ReSi] M. Reed, B. Simon. Methods of modern mathematical physics. Analysis
op-erators, Vol. 4. Academic Press, New York, 1979.
[Ri] L. Rienard. Spectral theory of Random Schr¨odinger operators. Springer–
Verlag, New York. 1990.
(Recibido en diciembre de 2003; la versi´on revisada en octubre de 2005)
Fatiha Gherib Laboratoire de physique quantique de la mati`ere et mod´elisations math´ematiques Centre universitaire de Mascara, Algeria e-mail: [email protected], fatiha [email protected] Bekkai Messirdi Faculty of Sciences, Department of Mathematics University of Oran, Algeria e-mail: [email protected], [email protected]
